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        <td class="header">&nbsp; Polarimetric Decomposition</td>
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<h3>Polarimetric Decomposition Operator</h3>&nbsp;&nbsp;&nbsp;This operator performs the following polarimetric
decompositions&nbsp;for a full polarimetric SAR product:<br>
<ul>
    <li>Sinclair Decomposition</li>
    <li>Pauli Decomposition</li>
    <li>Freeman-Durden Decomposition</li>
    <li>Yamaguchi Decomposition</li>
    <li>H-a Alpha Decomposition</li>
    <li>Touzi Decomposition</li>
    <li>Van Zyl Decomposition</li><li>Cloude Decomposition</li><li>Generalized&nbsp;Freeman-Durden Decomposition</li>
</ul>
<h4>Sinclair Decomposition</h4>&nbsp;&nbsp; Let&nbsp;<br>

<div style="margin-left: 40px;">&nbsp;&nbsp;&nbsp; <img style="width: 462px; height: 55px;" alt="" src="images/coherencyMatrixGenerationOp_eq1.jpeg"><br></div>
<br>&nbsp;&nbsp;&nbsp;
be the complex scatter matrix. The Sinclair decomposition produces (R,
G, B) bands with the following intensities<br>&nbsp;&nbsp;&nbsp; <br>
<ul>
    <li><span style="color: red;">Red</span>: |S<sub>vv</sub>|<sup>2</sup>,&nbsp;</li>
    <li><span style="color: rgb(0, 153, 0);">Green</span>: |(S<sub>hv</sub> + S<sub>vh</sub>)/2|<sup>2</sup>,&nbsp;</li>
    <li><span style="color: rgb(51, 51, 255);">Blue</span>: |S<sub>hh</sub>|<sup>2</sup>,&nbsp;</li>
</ul>
&nbsp;&nbsp;&nbsp; The main drawback of this decomposition is the physical interpretation of the resulting RGB
image.<br><h4>Pauli Decomposition</h4>&nbsp; &nbsp;The (R, G, B) bands produced by the Pauli decomposition correspond to
the following intensities [1]:<br>
<ul>
    <li><span style="color: red;">Red</span>: 0.5*|S<sub>hh</sub> - S<sub>vv</sub>|<sup>2</sup>,&nbsp;which represents
        the contribution of&nbsp;single- or odd-bounce scattering to the final measured scattering matrix
    </li>
    <li><span style="color: rgb(0, 153, 0);">Green</span>:&nbsp;0.5*|S<sub>hv</sub> + S<sub>vh</sub>|<sup>2</sup>,&nbsp;which
        represents the scatter power by targets that are able to return
        the orthogonal polarization, for example the volume scattering produced
        by the forest canopy
    </li>
    <li><span style="color: rgb(51, 51, 255);">Blue</span>:&nbsp;0.5*|S<sub>hh</sub> + S<sub>vv</sub>|<sup>2</sup>,&nbsp;which&nbsp;represents
        the power scattered&nbsp;by targets characterized by double- or even-bounce
    </li>
</ul>
<h4>Freeman-Durden Decomposition</h4>&nbsp; &nbsp;The Freeman decomposition models the covariance matrix as the
contribution of three scattering mechanisms [1]:<br>
<ol>
    <li>canopy scatter from a cloud of randomly oriented dipoles, forest for example;</li>
    <li>even- or double-bounce scatter from a pair of orthogonal surfaces with different dielectric constants;</li>
    <li>Bragg scatter from a moderately rough surface.</li>
</ol>
&nbsp;&nbsp;&nbsp;
The power scattered by the components of the above three scattering
mechanisms are employed to generate a RGB image as the following:<br>
<ul>
    <li><span style="color: red;">Red</span>: the power scattered by the double-bounce component of the covariance
        matrix
    </li>
    <li><span style="color: rgb(0, 153, 0);">Green</span>: the power scattered by the volume scattering component of the
        covariance matrix
    </li>
    <li><span style="color: rgb(51, 51, 255);">Blue</span>: the power scattered by the surface-like&nbsp;scattering
        component of the covariance matrix
    </li>
</ul>
<h4>Yamaguchi Decomposition</h4>
&nbsp;&nbsp; The three-component Freeman-Durden decomposition [1] can be
successfully applied to SAR observations under the reflection symmetry
assumption. However, there exists areas in an SAR image where the
reflection symmetry condition does not hold. Yamaguchi et al. proposed,
in 2005, a four-component scattering model by introducing an additional
term corresponding to nonreflection symmetric cases. The fourth
component introduced is equivalent to a helix scattering power. This
helix scattering power term appears in heterogeneous areas (complicated
shape targets or man-made structures) whereas disappears for almost all
natural distributed scattering. Therefore, Yamaguchi decomposition
models the covariance matrix as the following four scattering
mechanisms:<br>
<ol>
    <li>volume;</li>
    <li>double-bounce;</li>
    <li>surface; and</li>
    <li>helix scatter components.</li>
</ol>
<h4>H-A-Alpha Decomposition</h4>

&nbsp; &nbsp;The H-A-Alpha decomposition [1] is based on the eigen decomposition of the coherency matrix [T<sub>3</sub>].
Let <span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;; font-style: italic;">&#955;</span><sub style="font-style: italic;">1</sub>, <span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;; font-style: italic;">&#955;</span><sub style="font-style: italic;">2</sub>, and <span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;; font-style: italic;">&#955;</span><sub style="font-style: italic;">3</sub> be the eigenvalues of the coherency matrix (<span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;; font-style: italic;">&#955;</span><sub style="font-style: italic;">1</sub> <span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;">&gt; <span style="font-style: italic;">&#955;</span></span><sub style="font-style: italic;">2</sub> &gt; <span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;; font-style: italic;">&#955;</span><sub style="font-style: italic;">3</sub> &gt; 0), and <span style="font-style: italic;">u</span><sub style="font-style: italic;">1</sub>, <span style="font-style: italic;">u</span><sub style="font-style: italic;">2</sub>
and <span style="font-style: italic;">u</span><sub style="font-style: italic;">3</sub> be the corresponding eigenvectors
which can be expressed as the following:<br>

<div style="margin-left: 120px;"><img style="width: 325px; height: 41px;" alt="" src="images/polarimetricDecompositionOp_eq4.jpeg"><br>
</div>
<br>
&nbsp;Then three secondary parameters are defined as the follows:<br>
<ol>
    <li>Entropy:</li>
</ol>
<div style="margin-left: 120px;"><img style="width: 284px; height: 51px;" alt="" src="images/polarimetricDecompositionOp_eq1.jpeg"></div>
<ol>
    <li>Anistropy:&nbsp;</li>
</ol>
<div style="margin-left: 120px;"><img style="width: 97px; height: 51px;" alt="" src="images/polarimetricDecompositionOp_eq2.jpeg"></div>
<ol>
    <li>Alpha: <br>
    </li>
</ol>
<div style="margin-left: 120px;"><img style="width: 97px; height: 51px;" alt="" src="images/polarimetricDecompositionOp_eq3.jpeg"><br>
</div>
<h4>Touzi Decomposition</h4>

&nbsp;&nbsp; In 2007, for the monostatic scattering case, Ridha Touzi
has proposed in [2] a new Target Scattering Vector Model (TSVM). Based on
the Kennaugh-Huynen decomposition, this model allows to extract four
roll-invariant parameters:<br>
<ol>
    <li>Kennaugh-Huynen maximum polarization parameter: orientation angle (<span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;">&#936;</span>);
    </li>
    <li>Kennaugh-Huynen maximum polarization parameter: helicity (<span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;">&#964;</span>);
    </li>
    <li>Symmetric scattering type magnitude (<span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;">&#945;</span>);
    </li>
    <li>Symmetric scattering type phase (<span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;">&#934;</span>).<br>
    </li>
</ol>
&nbsp;&nbsp;&nbsp;
The roll-invariant incoherent target decomposition, i.e. Touzi decomposition, is as the following:<br>
<ol>

    <li>Compute target coherency matrix [T<sub>3</sub>] with a sliding window;</li>
    <li>Perform eigendecomposition on the coherency matrix;</li>
    <li>Apply the new target scattering vector model to each eigenvector to extract four parameters (<span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;">&#936;</span><sub>k</sub>, <span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;">&#964;</span><sub>k</sub>, <span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;">&#945;</span><sub>k</sub>, <span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;">&#934;</span><sub>k</sub>, k = 1, 2, 3).
    </li>
    <li>Compute averaged parameters (<span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;">&#936;</span>, <span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;">&#964;</span>, <span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;">&#945;</span>, <span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;">&#934;</span>): <br>
    </li>
</ol>
<div style="margin-left: 120px;"><img style="width: 393px; height: 46px;" alt="" src="images/polarimetricDecompositionOp_eq5.jpeg"></div>
<ol>

</ol>

<h4>Van Zyl Decomposition</h4>
&nbsp; &nbsp;The Van Zyl decomposition [1] assumes that the reflection
symmetry hypothesis establishes and the correlation between
co-polarized and cross-polarized channels is zero. The assumption is
generally true in case of natual media such as soil and forest. With
such an assumption, the eigen decomposition of the averaged covariance
matric C<sub>3</sub> can be given&nbsp; analytically&nbsp; and C<sub>3</sub> can be expressed in the following manner:
<br>
<img style="width: 357px; height: 84px;" alt="" src="images/polarimetricDecompositionOp_eq6.jpeg"><br>
<br>
The van Zyl decomposition thus shows that the first two eigenvectors
represent equivalent scattering matrices that can be interpreted in
terms of odd and even numbers of reflections.<br><h4>Cloude Decomposition</h4>
&nbsp; &nbsp;The Cloude decomposition [1] is an eigenvector based
decomposition. It idetifies the dominant scattering mechanism via the
extraction of the largest eigenvalue. <br><h4>Generalized&nbsp;Freeman-Durden Decomposition</h4>
&nbsp; &nbsp;The Generalized Freeman-Durden decomposition in [3] is
also a model-based decomposition. Similar to the Freeman-Durden
decomposition, it assumes that the main &nbsp;backscattering components
are direct backscatter from the underlying surface, double-bounce from
a pair of orthogonal surfaces, and direct volume scattering from the
top layer. With the 2-layer model and the assumptions above, we have
more parameters than the measurements. Generally, to invert the
modle,&nbsp;more assumptions on the parameters are needed.With the
Generalized Freeman-Durden decomposition, it is assumed that the
surface and dihedral mechanisms are orthogonal. Then the power&nbsp;
scattered by the three components can be computed by inverting the
model. <br>
<span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;; font-style: italic;"></span><span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;; font-style: italic;"></span><span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;; font-style: italic;"></span><span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;; font-style: italic;"></span><span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;"><span style="font-style: italic;"></span></span><span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;; font-style: italic;"></span><span style="font-style: italic;"></span><span style="font-style: italic;"></span><span style="font-style: italic;"></span>
<h4>Input and Output</h4>
<ul>
    <li>The
        input to this operator can be a full polarimetric SAR product with 8
        bands,
        i.e. I and Q bands for HH, VV, HV and VH polarizations, or covariance
        matrix generated by Covariance Matrix Generation operator, or coherency
        matrix output by Coherency Matrix Generation operator.
    </li>
    <li>The output of this operator are bands corresponding to the decomposition result.</li>
</ul>
<ol>
</ol>
<h4>Parameters Used</h4>&nbsp;&nbsp; For all decompositions, the following processing parameter is needed (see Figure
1):<br>
<ul>
    <li>Decomposition: the decomposition method</li>
</ul>
<br>
<img style="width: 500px; height: 500px;" alt="" src="images/polarimetricDecompositionOp.jpeg"><br>

<div style="text-align: left;">&nbsp;&nbsp;&nbsp;
    &nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;
    &nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;
    &nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;
    &nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;
    &nbsp;&nbsp;&nbsp; <br>&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;
    &nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp; &nbsp;Figure 1. Dialog box for
    Polarimetric Decomposition operator<br></div>
<br><br>
For Freeman-Durden decomposition, an extra parameter is needed (see Figure 2):<br>
<ul>
    <li>Window Size: dimension of sliding window for computing mean covariance or coherence matrix</li>
</ul>
<img style="width: 500px; height: 500px;" alt="" src="images/polarimetricDecompositionOp_Freeman.jpeg"><br><br>&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp; Figure 2. Dialog box for Freeman-Durden decomposition<br>
<br>
<br>
For Yamaguchi decomposition, the following parameters are needed (see Figure 3):<br>
<ul>
    <li>Window Size: dimension of sliding window for computing mean covariance or coherence matrix</li>
</ul>
<img style="width: 450px; height: 450px;" alt="" src="images/polarimetricDecompositionOp_Yamaguchi.jpeg"><br>
<br>
&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp; Figure 3. Dialog box for Yamaguchi decomposition<br>

<br>

For H-A-Alpha decomposition, the following extra parameters are needed (see Figure 4):<br>


<ul>
    <li>Window Size: dimension of sliding window for computing mean covariance or coherence matrix</li>
    <li>Checkbox for outputing parameters Entropy (H), Anistropy (A) and Alpha</li>
    <li>Checkbox for outputing parameters Beta, Delta, Gamma and Lambda</li>
    <li>Checkbox for outputing parameters Alpha1, Alpha2 and Alpha3</li>
    <li>Checkbox for outputing parameters Lambda1, Lambda2 and Lambda3</li>
</ul>


<img style="width: 500px; height: 500px;" alt="" src="images/polarimetricDecompositionOp_HAAlpha.jpeg"><br>
<br>
&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp; Figure 4. Dialog box for H-A-Alpha decomposition<br>

<br>
<br>
For Touzi decomposition, the following extra parameters are needed (see Figure 5):<br>

<ul>
    <li>Window Size: dimension of sliding window for computing mean covariance or coherence matrix</li>
    <li>Checkbox for outputing parameters Psi, Tau, Alpha and Phi</li>
    <li>Checkbox for outputing parameters Psi1, Tau1, Alpha1 and Phi1</li>
    <li>Checkbox for outputing parameters Psi2, Tau2, Alpha2 and Phi2</li>
    <li>Checkbox for outputing parameters Psi3, Tau3, Alpha3 and Phi3</li>
</ul>

<br>
<img style="width: 500px; height: 500px;" alt="" src="images/polarimetricDecompositionOp_Touzi.jpeg"><br>
<br>
&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp; Figure 5. Dialog box for Touzi decomposition<br>
<br>
For Van Zyl decomposition, the following parameters are used (see Figure 6):<br>
<ul>
    <li>Window Size: dimension of sliding window for computing mean covariance or coherence matrix</li>
</ul>
<img style="width: 450px; height: 450px;" alt="" src="images/polarimetricDecompositionOp_Van_Zyl.jpeg"><br>

<p><br>
</p>

<p>
    &nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;
    &nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp; Figure 6. Dialog box for Van Zyl decomposition</p>

<p><br>

</p>

<p> Reference:&nbsp;</p>

<p>[1] Jong-Sen Lee and Eric Pottier, Polarimetric Radar Imaging: From Basics to Applications, CRC Press, 2009</p>

<p>[2] R. Touzi, &#8220;Target Scattering Decomposition in Terms of
    Roll-Invariant Target Parameters,&#8221; IEEE Transactions on Geoscience and
    Remote Sensing, vol. 45, no. 1, pp. 73&#8211;84, January 2007.</p><p>[3] S.R. Cloude, &#8220;Polarisation: Applications in Remote Sensing&#8221;, Oxford University Press, ISBN 978-0-19-956973-1, 2009. </p><hr>
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